Painted Cubes

Task 160 ... Years 4 - 12

Summary

A cube made from unit cubes is spray painted on all faces, when deconstructed into its unit cubes, how many of these have 3, 2, 1, 0 faces painted. This classic problem is full of pattern and algebra investigations and yet the students don't have to be experienced algebraists to 'get it'. The generalisations may be developed from the tabulated data or they may be developed by visual inspection combined with knowledge of the number of faces, edges and vertices of a cube.

This cameo has a From The Classroom section which shows the journal work of a teacher in training challenged to explore a visual representation of the difference between two cubes as a result of exploring Painted Cubes.

Materials

3 wooden cubes of different sizes

board, marker & cloth

Content

multiplication calculations in context

number patterns

linear, quadratic & cubic functions

generalisation

symbolic representation

substituting into equations

solving equations

graphing ordered pairs

isometric drawing

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The opening question may seem a little straight forward, but its intent is to encourage students to see 'into' the scored cube - to deconstruct it. This visual separation of the whole into parts is important in developing the patterns in the main problem. If you have 2cm cubes (or similar) in the room it can be useful to ask students to show you how their calculation methods can be demonstrated. Two ways to calculate are:

3 layers, each with 9 unit cubes

9 towers each with 3 unit cubes

and there may be more. The background question here is the mathematician's question Can I check it another way? and as suggested by this extract from Maths300 ETuTE, Page 42, the exercise opens the door to further arithmetic in context.

Counting cubes gives the following lines of the table:

Large Cube

3

2

1

0

Total

3

8

12

6

1

27

4

8

24

24

8

64

5

8

36

54

27

125

6

...

...

...

...

216

There may be a Size 10 cube available similar to the ones in the task, but it is unlikely that you will be able to find the other sizes. If you have 2cm wooden cubes, or linking cubes such as Multilink the students could make the Size 6 - or perhaps enough of it to be able to calculate. Alternatively, or as well, students can record on isometric paper and colour code the unit cubes.

With one more line of data from the Size 6, patterns will start to occur to many students. But to tackle this problem as a number hunt is to miss some of its simple beauty - after all, wherever there is a number pattern there will be a corresponding visual pattern and vice versa. So perhaps looking at the geometry of the painted cube will help.

For any size cube (Size n):

Drawn by Becky & Lydia, Settlebeck High School
(see below)

The unit cubes with 3 painted faces will always be at the corners. There are 8 corners, so that column will always be 8.

The unit cubes with 2 faces painted will always be along the edges and will be two units less than the length of the edge because one edge cube is used for each corner. There are 12 edges on a cube so the second column will always be 12(n - 2).

The unit cubes with 1 face painted will always be squares in the middle of each face. The side of the square will be (n - 2) because each side is stops when it gets to an edge. There are 6 faces on a cube so the third column will always be 6(n - 2)2.

The unit cubes with 0 faces painted will always be a cube in the middle of the bigger cube. The side of the cube will be (n - 2) because in each direction it is stopped when it gets to a face of the original cube. So the fourth column will always be (n - 2)3.

With this insight, the remainder of the table can be filled in more easily.

Extension

The Size number can be paired with the corresponding number in any of the columns to make a set of ordered pairs. For example the ordered pairs for 2 painted faces are (3, 12), (4, 24), (5, 36) ...
For each column make a set of ordered pairs and graph them. What can you learn from these graphs?

Suppose you have 15,625 unit cubes. What size is the large cube? How many unit cubes have 3, 2, 1, 0 faces painted?

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

With enough wooden cubes or linking cubes and students working in groups the task can be converted to a whole class investigation. An advantage of linking cubes is that they can be joined to create just the 12 edges of a cube, thereby encouraging visualisation of the other parts that do and do not receive paint. The information above will guide the lesson.

John Hancock, Settlebeck High School, Cumbria, England, led his class on just such an investigation. Cube Tube includes two short videos taken in the first lesson of this investigation. Students recorded in their journals as they explored and when the class had discovered as much as was possible, John challenged them to publish their findings. Becky & Lydia's PowerPoint Report is an example of student publishing from Year 8. This link includes comment by John Hibbs (retired inspector of schools) following a visit to the school.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 38, Painted Cubes, which includes an Investigation Guide with answers and discussion and four more PowerPoint presentations from classmates of Becky and Lydia.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

Högskolan Malmö
Sweden

Shpetim Ademi
Teacher in training

Barbro Söderberg's class of teachers training to teach Gymnasium (Years 10 - 12) explored Painted Cubes. They were challenged to consider that if there is small cube inside a bigger cube then the material between the two must be the difference between the two cubes. They were encouraged to imagine slicing away sections of the outer cube to reveal the inner one and then to try rearranging the slices to create an expression for the difference.

Shpetim's shows how this tweak of Painted Cubes leads to serious high level algebra. (Click the image to reveal a larger version.)